Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

Abstract

A new hierarchy of Banach spaces $T_k(d,θ)$, $k$ any positive integer, is constructed using barriers in high dimensional Ellentuck spaces \cite{DobrinenJSL15} following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space \cite{Argyros/TodorcevicBK}. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of $\ell_\infty^n$, with the bound constant for all $n$. For each fixed pair $d$ and $θ$, the spaces $T_k(d,θ)$, $k\ge 1$, are $\ell_p$-saturated, forming natural extensions of the $\ell_p$ space, where $p$ satisfies $dθ=d^{1/p}$. Moreover, they form a strict hierarchy over the $\ell_p$ space: For any $j<k$, the space $T_j(d,θ)$ embeds isometrically into $T_k(d,θ)$ as a subspace which is non-isomorphic to $T_k(d,θ)$.